\(\int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx\) [554]

Optimal result

Integrand size = 18, antiderivative size = 98 \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=-\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{3 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]

[Out]

-2/3*x*csch(b*x+a)^(3/2)/b-4/3*cosh(b*x+a)*csch(b*x+a)^(1/2)/b^2+4/3*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(1/2)
/sin(1/2*I*a+1/4*Pi+1/2*I*b*x)*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))/b^2/csch(b*x+a)^(1/2)/(I*sinh(
b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5553, 3853, 3856, 2719} \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=-\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b} \]

[In]

Int[x*Cosh[a + b*x]*Csch[a + b*x]^(5/2),x]

[Out]

(-4*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(3*b^2) - (2*x*Csch[a + b*x]^(3/2))/(3*b) - (((4*I)/3)*EllipticE[(I*a -
Pi/2 + I*b*x)/2, 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 5553

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^(m -
n + 1))*(Csch[a + b*x^n]^(p - 1)/(b*n*(p - 1))), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csch[a + b
*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx}{3 b} \\ & = -\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{3 b} \\ & = -\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \sqrt {i \sinh (a+b x)} \, dx}{3 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = -\frac {4 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{3 b^2}-\frac {2 x \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{3 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\text {csch}(a+b x)} \left (2 \cosh (a+b x)+b x \text {csch}(a+b x)-2 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}\right )}{3 b^2} \]

[In]

Integrate[x*Cosh[a + b*x]*Csch[a + b*x]^(5/2),x]

[Out]

(-2*Sqrt[Csch[a + b*x]]*(2*Cosh[a + b*x] + b*x*Csch[a + b*x] - 2*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*S
qrt[I*Sinh[a + b*x]]))/(3*b^2)

Maple [F]

\[\int x \cosh \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )^{\frac {5}{2}}d x\]

[In]

int(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x)

[Out]

int(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=\text {Timed out} \]

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(5/2), x)

Giac [F]

\[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]

[In]

int(x*cosh(a + b*x)*(1/sinh(a + b*x))^(5/2),x)

[Out]

int(x*cosh(a + b*x)*(1/sinh(a + b*x))^(5/2), x)